Hopf invariant

In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between spheres.

Motivation

In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map

$\eta\colon S^3 \to S^2$,

and proved that $\eta$ is essential, i.e. not homotopic to the constant map, by using the linking number (=1) of the circles

$\eta^{-1}(x),\eta^{-1}(y) \subset S^3$ for any $x \neq y \in S^2$.

It was later shown that the homotopy group $\pi_3(S^2)$ is the infinite cyclic group generated by $\eta$. In 1951, Jean-Pierre Serre proved that the rational homotopy groups

$\pi_i(S^n) \otimes \mathbb{Q}$

for an odd-dimensional sphere ($n$ odd) are zero unless i = 0 or n. However, for an even-dimensional sphere (n even), there is one more bit of infinite cyclic homotopy in degree $2n-1$.

Definition

Let $\phi \colon S^{2n-1} \to S^n$ be a continuous map (assume $n>1$). Then we can form the cell complex

$C_\phi = S^n \cup_\phi D^{2n},$

where $D^{2n}$ is a $2n$-dimensional disc attached to $S^n$ via $\phi$. The cellular chain groups $C^*_\mathrm{cell}(C_\phi)$ are just freely generated on the $n$-cells in degree $n$, so they are $\mathbb{Z}$ in degree 0, $n$ and $2n$ and zero everywhere else. Cellular (co-)homology is the (co-)homology of this chain complex, and since all boundary homomorphisms must be zero (recall that $n>1$), the cohomology is

$H^i_\mathrm{cell}(C_\phi) = \begin{cases} \mathbb{Z} & i=0,n,2n, \\ 0 & \mbox{otherwise}. \end{cases}$

Denote the generators of the cohomology groups by

$H^n(C_\phi) = \langle\alpha\rangle$ and $H^{2n}(C_\phi) = \langle\beta\rangle.$

For dimensional reasons, all cup-products between those classes must be trivial apart from $\alpha \smile \alpha$. Thus, as a ring, the cohomology is

$H^*(C_\phi) = \mathbb{Z}[\alpha,\beta]/\langle \beta\smile\beta = \alpha\smile\beta = 0, \alpha\smile\alpha=h(\phi)\beta\rangle.$

The integer $h(\phi)$ is the Hopf invariant of the map $\phi$.

Properties

Theorem: $h\colon\pi_{2n-1}(S^n)\to\mathbb{Z}$ is a homomorphism. Moreover, if $n$ is even, $h$ maps onto $2\mathbb{Z}$.

The Hopf invariant is $1$ for the Hopf maps (where $n=1,2,4,8$, corresponding to the real division algebras $\mathbb{A}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$, respectively, and to the double cover $S(\mathbb{A}^2)\to\mathbb{PA}^1$ sending a direction on the sphere to the subspace it spans). It is a theorem, proved first by Frank Adams and subsequently by Michael Atiyah with methods of topological K-theory, that these are the only maps with Hopf invariant 1.

Generalisations for stable maps

A very general notion of the Hopf invariant can be defined, but it requires a certain amount of homotopy theoretic groundwork:

Let $V$ denote a vector space and $V^\infty$ its one-point compactification, i.e. $V \cong \mathbb{R}^k$ and

$V^\infty \cong S^k$ for some $k$.

If $(X,x_0)$ is any pointed space (as it is implicitly in the previous section), and if we take the point at infinity to be the basepoint of $V^\infty$, then we can form the wedge products

$V^\infty \wedge X$.

Now let

$F \colon V^\infty \wedge X \to V^\infty \wedge Y$

be a stable map, i.e. stable under the reduced suspension functor. The (stable) geometric Hopf invariant of $F$ is

$h(F) \in \{X, Y \wedge Y\}_{\mathbb{Z}_2}$,

an element of the stable $\mathbb{Z}_2$-equivariant homotopy group of maps from $X$ to $Y \wedge Y$. Here "stable" means "stable under suspension", i.e. the direct limit over $V$ (or $k$, if you will) of the ordinary, equivariant homotopy groups; and the $\mathbb{Z}_2$-action is the trivial action on $X$ and the flipping of the two factors on $Y \wedge Y$. If we let

$\Delta_X \colon X \to X \wedge X$

denote the canonical diagonal map and $I$ the identity, then the Hopf invariant is defined by the following:

$h(F) := (F \wedge F) (I \wedge \Delta_X) - (I \wedge \Delta_Y) (I \wedge F).$

This map is initially a map from

$V^\infty \wedge V^\infty \wedge X$ to $V^\infty \wedge V^\infty \wedge Y \wedge Y$,

but under the direct limit it becomes the advertised element of the stable homotopy $\mathbb{Z}_2$-equivariant group of maps. There exists also an unstable version of the Hopf invariant $h_V(F)$, for which one must keep track of the vector space $V$.